mpri-funcprog-project/coq/CPSIndifference.v

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Require Import MyTactics.
Require Import Sequences.
Require Import LambdaCalculusSyntax.
Require Import LambdaCalculusValues.
Require Import LambdaCalculusReduction.
Require Import CPSDefinition.
(* In a CPS term (i.e., a term produced by the CPS translation), the
right-hand side of every application is a value, and the left-hand
side of every [let] construct is a value. *)
Inductive is_cps : term -> Prop :=
| IsCPSVar:
forall x,
is_cps (Var x)
| IsCPSLam:
forall t,
is_cps t ->
is_cps (Lam t)
| IsCPSApp:
forall t1 t2,
is_cps t1 ->
is_cps t2 ->
is_value t2 ->
is_cps (App t1 t2)
| IsCPSLet:
forall t1 t2,
is_cps t1 ->
is_cps t2 ->
is_value t1 ->
is_cps (Let t1 t2)
.
(* To prove that the above invariant holds, we must also define what it means
for a continuation [c] to satisfy this invariant. *)
Inductive is_cps_continuation : continuation -> Prop :=
| IsCPSO:
forall k,
is_value k ->
is_cps k ->
is_cps_continuation (O k)
| IsCPSM:
forall K,
is_cps K ->
is_cps_continuation (M K).
Local Hint Constructors is_cps is_cps_continuation.
(* [is_cps] is preserved by renamings. *)
Lemma is_cps_renaming:
forall t,
is_cps t ->
forall sigma,
is_ren sigma ->
is_cps t.[sigma].
Proof.
induction 1; intros sigma Hsigma; asimpl;
try solve [ econstructor; obvious ].
(* Var *)
{ destruct Hsigma as [ xi ? ]. subst sigma. asimpl. econstructor. }
Qed.
Local Hint Resolve is_cps_renaming.
Lemma is_cps_continuation_renaming:
forall c i,
is_cps_continuation c ->
is_cps_continuation (liftc i c).
Proof.
induction 1; simpl; econstructor; obvious.
Qed.
Local Hint Resolve is_cps_continuation_renaming.
(* [is_cps] is preserved by substitution. *)
Lemma is_cps_substitution_aux:
forall sigma,
(forall x, is_cps (sigma x)) ->
(forall x, is_cps (up sigma x)).
Proof.
intros sigma H [|x]; asimpl.
{ econstructor. }
{ eapply is_cps_renaming; obvious. }
Qed.
Lemma is_cps_substitution:
forall K,
is_cps K ->
forall sigma,
(forall x, is_cps (sigma x)) ->
is_value_subst sigma ->
is_cps K.[sigma].
Proof.
induction 1; intros; asimpl; eauto;
econstructor; eauto using is_cps_substitution_aux with obvious.
Qed.
Lemma is_cps_substitution_0:
forall K v,
is_cps K ->
is_cps v ->
is_value v ->
is_cps K.[v/].
Proof.
intros. eapply is_cps_substitution; obvious.
intros [|x]; asimpl; eauto.
Qed.
(* Inversion lemmas for [is_cps]. *)
Lemma is_cps_Lam_inversion:
forall t,
is_cps (Lam t) ->
is_cps t.
Proof.
inversion 1; eauto.
Qed.
(* A CPS term reduces in the same manner in call-by-value and in call-by-name.
Thus, the CPS transformation produces terms that are "indifferent" to which
of these two reduction strategies is chosen. *)
Lemma cps_indifference_1:
forall t1, is_cps t1 ->
forall t2, cbv t1 t2 -> cbn t1 t2.
Proof.
induction 1; intros; invert_cbv; obvious.
Qed.
Lemma cps_indifference_2:
forall t1, is_cps t1 ->
forall t2, cbn t1 t2 -> cbv t1 t2.
Proof.
induction 1; intros; invert_cbn; obvious.
Qed.
(* [is_cps] is preserved by call-by-value and call-by-name reduction. *)
Lemma is_cps_cbv:
forall t,
is_cps t ->
forall t',
cbv t t' ->
is_cps t'.
Proof.
induction 1; intros; invert_cbv;
eauto using is_cps, is_cps_substitution_0, is_cps_Lam_inversion.
Qed.
Lemma is_cps_cbn:
forall t,
is_cps t ->
forall t',
cbn t t' ->
is_cps t'.
Proof.
induction 1; intros; invert_cbn;
eauto using is_cps, is_cps_substitution_0, is_cps_Lam_inversion.
Qed.
(* A CPS term reduces in the same manner in call-by-value and in call-by-name.
The statement is here generalized to a sequence of reduction steps. *)
Lemma cps_star_indifference_1:
forall t1 t2,
star cbv t1 t2 ->
is_cps t1 ->
star cbn t1 t2.
Proof.
induction 1; intros;
eauto using cps_indifference_1, is_cps_cbv with sequences.
Qed.
Lemma cps_star_indifference_2:
forall t1 t2,
star cbn t1 t2 ->
is_cps t1 ->
star cbv t1 t2.
Proof.
induction 1; intros;
eauto using cps_indifference_2, is_cps_cbn with sequences.
Qed.
(* The main auxiliary lemmas. *)
Lemma is_cps_apply:
forall c v,
is_cps_continuation c ->
is_cps v ->
is_value v ->
is_cps (apply c v).
Proof.
inversion 1; intros; simpl; eauto using is_cps_substitution_0.
Qed.
Lemma is_cps_reify:
forall c,
is_cps_continuation c ->
is_cps (reify c).
Proof.
inversion 1; simpl; eauto.
Qed.
Lemma is_value_reify:
forall c,
is_cps_continuation c ->
is_value (reify c).
Proof.
inversion 1; simpl; eauto.
Qed.
Local Hint Resolve is_cps_apply is_cps_reify is_value_reify.
(* The main lemma. *)
Lemma cps_form:
(
forall v,
is_value v ->
is_cps (cpsv v)
) /\ (
forall t c,
is_cps_continuation c ->
is_cps (cps t c)
).
Proof.
eapply mutual_induction.
(* [cpsv] *)
{ intros n IHcps v Hvn ?.
destruct v; [ | | false; obvious | false; obvious ].
{ cpsv; eauto. }
{ cpsv; eauto 6 with size. }
}
(* [cps] *)
{ intros n IHcpsv IHcps t c Htn Hc.
value_or_app_or_let t; cps.
(* Case: [t] is a value. *)
{ obvious. }
(* Case: [t] is an application. *)
{ eapply IHcps; [ size | econstructor ].
eapply IHcps; [ size | econstructor ].
econstructor; obvious. }
(* Case: [t] is a [let] construct. *)
{ eauto 8 with obvious. }
}
Qed.
Lemma cps_form_main:
forall t,
is_cps (cpsinit t).
Proof.
simpl. intros. eapply cps_form. unfold init. obvious.
Qed.
(* One property of CPS terms that we do not prove is that all applications are
in tail position, or, in other words, that there is no need for reduction
under a context. In fact, because a CPS-translated function expects two
arguments, there *is* a need for reduction under a context, but only under
a context of depth zero or one. *)